There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{J}}{\mathbf{J}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{M}}{\mathbf{M}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ PQ
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{RS}}{\mathbf{TU}}$.

There are two bags, A and B. Bag A contains four white balls and one red ball, and bag B contains two white balls and three red balls. Two balls are taken simultaneously out of bag A, then two balls are taken simultaneously out of bag B.
(1) The probability that two white balls are taken out of A, and one white ball and one red ball are taken out of B is $\frac{\mathbf{O}}{\mathbf{PQ}}$.
(2) The probability that the four balls taken out consist of three white balls and one red ball is $\frac{\mathbf{R}}{\mathbf{S}}$.
(3) The probability that the four balls taken out all have the same color is $\square$ T UV
(4) The probability that of the four balls taken out, two or fewer are white balls is $\frac{\mathbf{WX}}{\mathbf{W}}$.

On a coordinate plane, 12 points are arranged as shown in the figure to the right. If we are to select three points as vertices of a triangle, how many triangles are possible in total?
First, there are $\mathbf { K L M }$ ways to select three points from the 12 points.
Next, let us find how many ways it is possible to select three or more points in a straight line.
Let us look at the two cases.
(i) There are $\mathbf { N }$ straight lines that pass through four points.
(ii) There are $\mathbf { O }$ straight lines that pass through three points.
Hence, among all combinations of three points that are in a straight line and so cannot be the vertices of a triangle, $\mathbf { P Q }$ combinations belong to case (i), and $\mathbf { Q }$ combinations belong to case (ii).
Thus, the total number of possible triangles is $\mathbf{STU}$.
In particular, if we set $( 1,1 )$ as point A and $( 4,1 )$ as point B , then $\mathbf { V W }$ triangles have two vertices on segment AB.

On a coordinate plane, 12 points are arranged as shown in the figure to the right. If we are to select three points as vertices of a triangle, how many triangles are possible in total?
First, there are $\mathbf { K L M }$ ways to select three points from the 12 points.
Next, let us find how many ways it is possible to select three or more points in a straight line.
Let us look at the two cases.
(i) There are $\mathbf { N }$ straight lines that pass through four points.
(ii) There are $\mathbf { O }$ straight lines that pass through three points.
Hence, among all combinations of three points that are in a straight line and so cannot be the vertices of a triangle, $\mathbf { P Q }$ combinations belong to case (i), and $\mathbf { Q }$ combinations belong to case (ii).
Thus, the total number of possible triangles is $\mathbf{STU}$.
In particular, if we set $( 1,1 )$ as point A and $( 4,1 )$ as point B , then $\mathbf { V W }$ triangles have two vertices on segment AB.

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

In a box there are ten cards on which the numbers from 0 to 9 have been written successively. We take three cards out of the box using two methods and consider the probabilities.
(1) We take out three cards simultaneously.
(i) The probability that each number on the three cards is 2 or more and 6 or less is $\dfrac{\mathbf{KL}}{\mathbf{MN}}$.
(ii) The probability that the smallest number is 2 or less and the greatest number is 8 or more is $\dfrac { \mathbf { N O } } { \mathbf { P Q } }$.
(2) Three times we take out one card from the box, check its number, and then return it to the box. The probability that the smallest number is 2 or more and the greatest number is 6 or less is $\dfrac { \mathbf { R } } { \mathbf { S } }$.

A graduating class has 8 students responsible for planning a class trip, divided into three groups A, B, and C, consisting of 3, 3, and 2 people respectively. Each of the 8 students will be assigned to one of the groups, and two students, A and B, must be in the same group. How many ways are there to divide the 8 students into groups?
(1) 140 ways
(2) 150 ways
(3) 160 ways
(4) 170 ways
(5) 180 ways

From the nine numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, any three distinct numbers are randomly selected, with each number having equal probability of being selected. The probability that the product of the three numbers is a perfect square is (20). (Express as a fraction in lowest terms)

A convenience store packages three building block models (A, B, C) and five character figurines ($a, b, c, d, e$), totaling eight toys, into two bags for sale. Each bag contains four toys, and the packaging follows these principles: (I) A and $a$ must be in the same bag. (II) Each bag must contain at least one building block model. (III) $d$ and $e$ must be in different bags. Based on the above, select the correct options.
(1) Each bag must contain at least two character figurines
(2) B and C must be in different bags
(3) If B and $d$ are in the same bag, then C and $e$ must be in the same bag
(4) If B and $d$ are in different bags, then $b$ and $c$ must be in different bags
(5) If $b$ and $c$ are in different bags, then B and C must be in the same bag

An ice cream shop needs to prepare at least $n$ buckets of different flavors of ice cream to satisfy the advertisement claim that ``the number of combinations of selecting two scoops of different flavors exceeds 100 types.'' How many ways can a customer select two scoops (which may be the same flavor) from $n$ buckets?
(1) 101
(2) 105
(3) 115
(4) 120
(5) 225

Eight buildings are arranged in a row, numbered 1, 2, 3, 4, 5, 6, 7, 8 from left to right. A telecommunications company wants to select three of these buildings' rooftops to install telecommunications base stations. If base stations cannot be installed on two adjacent buildings to avoid signal interference, how many ways are there to select locations for the base stations if no base station is installed on building 3?
(1) 12
(2) 13
(3) 20
(4) 30
(5) 35

A bag contains blue, green, and yellow balls totaling 10 balls. Two balls are randomly drawn from the bag (each ball has an equal probability of being drawn). The probability that both balls drawn are blue is $\frac{1}{15}$, and the probability that both are green is $\frac{2}{9}$. The probability that two randomly drawn balls are of different colors is $\frac{\text{(16--1)}}{\text{(16--3)}}$. (Express as a fraction in lowest terms)

Divide the 50 positive integers from 1 to 50 equally into groups A and B, with 25 numbers in each group, such that the median of group A is 1 less than the median of group B. How many ways are there to divide them?
(1) $C_{25}^{50}$
(2) $C_{24}^{48}$
(3) $C_{12}^{24}$
(4) $\left(C_{12}^{24}\right)^{2}$
(5) $C_{24}^{48} \cdot C_{12}^{24}$

In the Elements of Geometry, it is stated: "Two distinct points determine a line." In general, three distinct points determine $C_{2}^{3} = 3$ lines; however, if these three points are collinear, only one line is determined. On the coordinate plane, circle $\Gamma_{1}: x^{2} + y^{2} = 4$ intersects the two coordinate axes at 4 points, circle $\Gamma_{2}: x^{2} + y^{2} = 2$ intersects the line $x - y = 0$ at 2 points, and circle $\Gamma_{2}$ intersects the line $x + y = 0$ at 2 points. How many different lines can these 8 points determine?
(1) 12
(2) 16
(3) 20
(4) 24
(5) 28

Answer the following questions.
(1) Calculate the number of possible ways to distribute $n$ equivalent balls to $r$ distinguishable boxes such that each box contains at least one ball, where $n \geq 1$ and $1 \leq r \leq n$.
Next, consider to place $n$ black balls and $m$ white balls in a line uniformly at random. A run is defined to be a succession of the same color. Let $r$ be the number of runs of black balls and $s$ be the number of runs of white balls. Assume that $n \geq 1 , m \geq 1,1 \leq r \leq n$, and $1 \leq s \leq m$.
(2) Calculate the total number of arrangements when we do not distinguish among balls of the same color.
(3) Calculate the probability $P ( r , s )$ that the number of runs of black balls is $r$ and the number of runs of white balls is $s$.
(4) Calculate the probability $P ( r )$ that the number of runs of black balls is $r$.
(5) Using $( 1 + x ) ^ { n } ( 1 + x ) ^ { m } = ( 1 + x ) ^ { n + m }$, show that the following equations hold.
$$\begin{aligned} \sum _ { \ell = 0 } ^ { \min \{ n , m \} } \binom { n } { \ell } \binom { m } { \ell } & = \binom { n + m } { m } \\ \sum _ { \ell = 0 } ^ { \min \{ n - 1 , m \} } \binom { n } { \ell + 1 } \binom { m } { \ell } & = \binom { n + m } { m + 1 } \end{aligned}$$
(6) Calculate the expected value $E ( r )$ and the variance $V ( r )$ of $r$.
Calculate $\lim _ { N \rightarrow \infty } \frac { E ( r ) } { N }$ and $\lim _ { N \rightarrow \infty } \frac { V ( r ) } { N }$ supposing that $N = n + m$ and $\lim _ { N \rightarrow \infty } \frac { n } { N } = \lambda$, where $\lambda$ is a real constant.

A bag contains 2 red, 2 white, and 1 yellow marble.
When 4 marbles are randomly drawn from the bag, what is the probability that the remaining marble in the bag is red?
A) $\frac { 1 } { 2 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 3 } { 5 }$

From a group of 6 girls and 7 boys, 2 representatives are selected.
What is the probability that one of the two selected representatives is a girl and the other is a boy?
A) $\frac { 3 } { 4 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 2 } { 13 }$
D) $\frac { 7 } { 13 }$
E) $\frac { 9 } { 13 }$

A florist has roses of 5 different colors in large quantities and 2 types of vases. A customer wants to buy a total of 3 roses of 2 different colors and 1 vase.
In how many different ways can this customer make the purchase?
A) 15
B) 20
C) 25
D) 40
E) 50

A bag contains 5 red and 4 white marbles.
When 3 marbles are drawn randomly from this bag at the same time, what is the probability that there are at most 2 marbles of each color?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 7 } { 8 }$
E) $\frac { 8 } { 9 }$

All 5 different marbles are to be distributed among 3 siblings of different ages.
In how many different ways can this distribution be made such that the oldest sibling gets 1 marble and the other two each get at least one marble?
A) 45
B) 50
C) 60
D) 70
E) 75

Four distinct marbles will be distributed to 3 siblings such that each sibling receives at least 1 marble.
In how many different ways can this distribution be done?
A) 24
B) 32
C) 36
D) 40
E) 48

Let $a , b , c$ be real numbers and $0 < b < 1$ such that
$$\begin{aligned} & a = b \cdot c \\ & a + c = b \end{aligned}$$
Given this, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$

Four identical matches are taken, each with only one flammable end. These matches are randomly arranged along all sides of a square whose side length is the same as the length of one match, with the ends touching each other.
What is the probability that there are no flammable ends in contact with each other in this arrangement?
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 3 } { 8 }$
D) $\frac { 1 } { 16 }$
E) $\frac { 3 } { 16 }$

$$\mathrm { X } \subseteq \{ \mathrm { a } , \mathrm {~b} , \mathrm { c } , \mathrm {~d} , \mathrm { e } \}$$
Given that, how many different subsets $X$ are there such that the number of elements in $\mathbf { X } \cap \{ \mathbf { a } , \mathbf { b } \}$ is 1?
A) 10 B) 12 C) 14 D) 16 E) 18

A school's basketball team has a total of 8 players, two of whom are brothers. 5 of these players will be selected to be in the starting lineup.
In how many different ways can a selection be made such that both brothers are in this lineup?
A) 20 B) 24 C) 30 D) 36 E) 40